For the following function, find the values of the constants π and π for which the function is continuous, but not differentiable.
π(π₯)={ππ₯+π,π₯>0 & sinπ₯,π₯β€0.
as with the previous problem, you want
f(0) to match up,
but f'(0) has a break.
how about a=0, b=0
or any linear function f(x) = kx where kβ 1
since f'(sinx) = 1 at x=0
To find the values of the constants π and π for which the function π(π₯) is continuous but not differentiable, we need to consider the conditions for continuity and differentiability separately.
For a function to be continuous, it needs to be defined at a point and must not have any gaps, jumps, or holes. In this case, π(π₯) is defined differently for π₯ > 0 and π₯ β€ 0.
- For π₯ > 0, π(π₯) = ππ₯ + π. To ensure continuity, we need the left-hand limit to be equal to the right-hand limit when π₯ approaches 0 from the left. This means we need to evaluate ππ₯ + π as π₯ approaches 0 from the left. Setting π₯ = 0 in the expression ππ₯ + π gives us π. Therefore, π needs to be equal to the value of the function at π₯ = 0, which is π(0) = sin(0) = 0. Hence, π = 0.
- For π₯ β€ 0, π(π₯) = sinπ₯. Since sinπ₯ is a well-defined function for all values of π₯, we don't need to consider any conditions for π or π.
Therefore, the values of π and π for which the function π(π₯) = ππ₯ + π is continuous are π = any value and π = 0.
Now let's discuss the differentiability of π(π₯). For a function to be differentiable at a point, it needs to have a well-defined derivative at that point, which means that the derivative from the left-hand side should be equal to the derivative from the right-hand side.
Looking at π(π₯), we observe that the derivative of π(π₯) = ππ₯ + π with respect to π₯ is π. This derivative is a constant and does not depend on π₯. On the other hand, when π₯ β€ 0, the derivative of π(π₯) = sinπ₯ with respect to π₯ is cosπ₯, which is defined for all π₯.
Since the derivative of π(π₯) = ππ₯ + π is a constant (π) and the derivative of π(π₯) = sinπ₯ is defined for all values of π₯, the function π(π₯) is differentiable for any values of π and π.
Therefore, the values of π and π for which the function π(π₯) is continuous but not differentiable are π = any value and π = 0.